Question

Rozwiąż równanie
cos2x=cosx

Answer 1

Cos2x = cosx
cos2x = 2cos²x - 1
2cos²x - 1 = cosx
2cos²x - cosx - 1 = 0
cosx = t ∈ <-1; 1>
2t² - t - 1 = 0
Δ = (-1)² - 4 × 2 × (-1) = 1 + 8 = 9
√Δ = 3
t₁ = (1 - 3) / 2×2 = -2/4 = -1/2
t₂ = (1 + 3) / 2×2 = 4/4 = 1
cosx = -1/2  ∨  cosx = 1
x₁ = 120° + k × 360°
x₂ = 330° + k × 360°
x₃ = k × 360°